
\paragraph{Precise structural quantities: $V_L$ and $D$}
Measurements of total lipid volume $V_L$ have been performed using a variety of techniques. Agreement between the different methods is about 3 parts in 1000 and the errors in each method alone is of order 2 parts in 1000~\cite{nagle00a}. The lipid volume of DMPC was first measured by~\cite{nagle78}, and remarkably that value is still valid.
An important experimental result is that the total lipid volume does not change measurably with hydration~\cite{nagle00a}.
\paragraph{Area per lipid $A_L$}
Knowledge of surface \index{area per lipid} can be correlated with many of the membrane structural and physical properties (e.g., order parameters, bilayer thickness, conformation of acyl chains, etc.) and hence this is considered to be the most central structural quantity~\cite{sanka04}.
\cite{kucer05a} obtained a result for the area of DMPC about 1\,\AA$^2$ larger than the earlier value (59.6\,A$^2$) reported by~\cite{nagle00a} (that was obtained by x-ray method~\cite{petra98a}, and by NMR~\cite{koeni97a, petra00a}). Although agreement is satisfactory within estimated uncertainties of 0.5\,\AA$^2$, the refined structure used by~\cite{kucer05a} was obtained with much better x-ray data, so the new value of $A_L=60.6\,$\AA$^2$ should be more accurate.

\paragraph{Bilayer thickness}
The bilayer thickness \index{thickness} is defined as the head-head separation $D_{HH}$~\cite{nagle00a}.

\subsection{Electron density profile}\index{electron density profile - EDP}

The electron density profile (EDP)
We can now compare the average from quartets of segmental order parameters $S^k_\textrm{mol}$ to the corresponding order parameters of our coarse-grain units.

The second order Legendre polynomial $P_2$ is widely used:
\[P_2=\frac{1}{2}(3\cos^2\theta-1)\,,\]
where $\theta$ is the angle between the direction of the bond and the bilayer normal.
\begin{equation}
S_{CD}^{(i)}=
\left<
\frac{1}{2}
\left(
3{cos^{2}}\beta_{i}-1
\right)
\right>
=
\left<
P_{2}(cos{\beta_{i}})
\right>.
\label{scd}
\end{equation}
Full alignment to the bilayer normal (i.e. $\theta=0$) is indicated by $P_2=1$, full order perpendicular to the bilayer normal with $P_2=-0.5$ and a random orientation with $P_2=0$.
For a perfectly ordered system, $\theta=0$ and the order parameter takes its maximum value, i.e. $P_2=1$; for a system perfectly ordered perpendicularly to the director.
For a uniform distribution, $P_2=0$. Experimentally, $P_2\approx0.2$ for the top and middle parts of the chains of DPPC, consistent with the partial order~\cite{pas02}. 
\begin{figure}%[ht]
\centering
\includegraphics[scale=1]{biophys/douliezAxesSystemsEdited}
\caption[Phospholipid intermolecular and intramolecular motions]{Schematic representation of a phospholipid molecule with the axes systems needed for the description of the intermolacular and intramolecular motions. $Z_N$ and $Z_D$ represent the bilayer normal and the diffusion axis of the molecule, $S_k^{CD}$ and $S_k^{CC}$ the $C_k-D$ and the $C_k-C_{k-1}$ bond order parameters, $l_k$ and $L_k$ the average segment length projected onto the diffusion axis and onto the bilayer normal, respectively. $\theta_{inter}$ accounts for intermolecular averages, and $\psi_{isom}$ and $\theta_{isom}$ account for the intramolecular motions. Adapted from Douliez et al.~\cite{douliez95}}
\label{fig:stevBeadSpring}
\end{figure}   

\subsubsection{Compressibility moduli}
The response of the bilayer surface area $A$ to an isotropic tension $\overline{T}$ is specified by the isothermal modulus of the surface compressibility:
\begin{equation}
K_A= A \left(\frac{\partial \overline{T}}{\partial A}\right)_T
\end{equation} 
The area compressibility modulus~$K_A$ is alternatively defined as the energy~$E_{K_A}$ per unit area to spend in order to stretch an interface $A_0$ to produce an area change $\Delta A$ according to Hooke's law: 
\begin{equation}
E_{K_A}=\frac{1}{2}K_A\left(\frac{\Delta A}{A_0}\right)^2 
\end{equation}
$K_A$ describes an elastic property of the membrane and can be related to the variance of the lipid area $\sigma^2_{A}$:
\begin{equation}
K_A=\frac{k_BTA}{N_l\sigma^2_{A}} 
\end{equation}
with $A$ the average area per lipid, $N_l$ the number of lipid molecules per layer, $T$ the temperature and $k_B$ the Boltzmann constant. %~\cite{anezo03a}.
 
\begin{equation}
K_A=A\,\frac{\partial\gamma}{\partial A} 
\end{equation}

%The experimentalists~\cite{petra98a} have estimated the area compressibility of fully-hydrated DMPC at 30\textcelsius: K$_A=108\pm35$\,dyn/cm.

\begin{table}[]
\begin{center}
\vspace{8pt}
\begin{tabular}{lcccc} % put @{} if you want to eliminate horizontal space between columns
%Stretch modulus
&\cite{evans90}& \cite{koeni97a} & \cite{petra98a}&\cite{rawicz00} \\ 
K$_A$\,[dyn/cm] &$145\pm10$ &$137\pm15$ &  $108\pm35$ &$234\pm23$ 
\end{tabular}
\caption[Area compressibility modulus]{Fluid-phase DMPC area compressibility modulus. Experimental measurements of~$K_A$ reported in the literature.}
\label{tab:structData}
\end{center}
\end{table}

\cite{ipsen90}


\subsection{Volume compressibility modulus}\index{elasticity!volume compressibility modulus $K_V$}
The volume (or bulk) compressibility modulus~$K_V$ is defined as~\cite[Chapter~12]{cevc}: %p.348
\begin{equation}
K_V=-V\left(\frac{\partial P}{\partial V}\right)_T 
\end{equation}
$K_V$ describes the response of the bilayer volume $V$ to uniform hydrostatic pressure $P$.
Typical values of the bulk modulus are $K_V\sim10-30$\,kbar for fluid lipid bilayers~\cite[Chapter~12]{cevc}. These are in the range found for normal incompressible fluids, in contrast to the considerable greater surface compressibility of the bilayer.


Experimental measurements of the bending modulus for fluid-phase DMPC bilayers are reported in Table~\ref{tab:bendMod}.
\begin{table}[]
\begin{center}
\vspace{8pt}
\begin{tabular}{cccc} % put @{} if you want to eliminate horizontal space between columns
  $\kappa$ / $10^{-19}$J    & $\kappa$ / $k_BT$ & T / $^{\circ}{\rm C}$ & {\em experiment}\\ 
 1.30 & & 30& shape fluctuation~\cite{meleard97}\\
 0.56 & 14 &  29 &  pipette aspiration~\cite{rawicz00}\\ 
 1.33 & 33  &  30 &  all-optical~\cite{lee01} \\
 0.69 & 17  &  30 &  x-ray scattering~\cite{chu05} 
\end{tabular}
\caption[Bending modulus of fluid-phase DMPC bilayers]{Bending modulus $\kappa$ of fluid-phase DMPC bilayers.}
\label{tab:bendMod}
\end{center}
\end{table}
There are clear discrepancies in the data obtained by different groups using different techniques. 
\begin{table}[]
\begin{center}
\vspace{8pt} 
\begin{tabular}{lll} % put @{} if you want to eliminate horizontal space between columns
     & $k$ / $k_BT$ & $c_0$ / nm$^{-1}$\\ 
DMPC$_{\,29^{\circ}{\rm C}}$ & 7~\cite{rawicz00} & \\
DMPC$_{\,30^{\circ}{\rm C}}$ & 16~\cite{lee01} & \\
DOPC$_{\,18^{\circ}{\rm C}}$ & 10~\cite{rawicz00} & \\
DOPC$_{\,32^{\circ}{\rm C}}$ &  & $-0.114$~\cite{chen97} \\
DOPC$_{\,30^{\circ}{\rm C}}$ &  & $-0.112$~\cite{keller93} \\
DOPE$_{\,30^{\circ}{\rm C}}$ &  & $-0.526$~\cite{keller93} 
%DOPE/tetradecane$_{\,22^{\circ}{\rm C}}$ &  & $-0.340$~\cite{chen97} \\
\end{tabular}
\caption[Curvature elastic and geometric data]{Curvature elastic and geometric data of fluid-phase monolayers.}
\label{tab:curvData}
\end{center}
\end{table}

\subsection{Lipid diffusion}\index{diffusion!lipid!experimental data}

The average distance $s$ traversed in time $t$ depends on $D$ according to the expression: %http://www.ncbi.nlm.nih.gov/books/bv.fcgi?rid=stryer.section.1687

\begin{equation}\label{eq:diffDist}
s = \sqrt{4Dt}
\end{equation}

The diffusion coefficient of lipids in a variety of membranes is about 1\,$\mu$m$^2$/s. Thus, a phospholipid molecule diffuses an average distance of 2\,$\mu$m in 1\,s. This rate means that a lipid molecule can travel from one end of a bacterium to the other in a second. The magnitude of the observed diffusion coefficient indicates that the viscosity of the membrane is about 100 times that of water, rather like that of olive oil~\cite{berg}.

Lipid lateral diffusion can be measured using the pulsed field gradient NMR method~\cite{oradd02, filippov03}: this method is non-perturbing and in most cases does not need any labeling. 

Experimental data for DMPC at $30^{\circ}{\rm C}$ are collected in Table~\ref{tab:diffData}. 
\begin{table}[]
\begin{center}
\vspace{8pt}
\begin{tabular}{lccc} % put @{} if you want to eliminate horizontal space between columns
Experiment & NMR~\cite{filippov03} & NMR~\cite{oradd02}& Fluorescence recovery~\cite{almeida92}% &FRAP~\cite{blume93}&Excimer~\cite{blume93} %values from Hogberg JCPB 110 
\\Diffusion [nm$^2$/$\mu$s] & 9 & 9 & 6 %&$2-8\times10^{-8}$&2-30$\times10^{-8}$
\end{tabular}
\caption[DMPC experimental diffusion data]{DMPC experimental diffusion data, full hydration, $30^{\circ}{\rm C}$.}
\label{tab:diffData}
\end{center}
\end{table}
Considering the diffusion for fluid DMPC, we compute an average travelling distance of $6$\,nm per $\mu$s. 

\cite{haibel98} estimated the translational diffusion from the orientational relaxation of the lipid headgroup in alternating electric fields; the diffusion coefficients are about one order of magnitudes larger than FRAP and NMR measurements.

Free-volume diffusion theory~\cite{cohen59, turnbull61, turnbull70}.

Lipid diffusion is assumed to proceed by ``hopping'' of molecules into vacancies formed by lateral-density fluctuations~\cite{oleary87}.

For diffusion coefficients of the order of $D\sim1\,$nm$^2/\mu$s, \cite{vaz91} predicted that each lipid performs a ``jump'' every 160\,ns, on average.


\subsubsection{Curvature elasticity theory}

According to the general theory developed by Helfrich,~\cite{helfrich73,sedd95,marsh06}  the surface curvature elastic energy per unit area $g$ is expressed as: 
\begin{equation}\label{eq:helfrich}
g = \kappa\left(c_1 + c_2 -c_0\right)^2/2 + \kappa_\textrm{G}\, c_1\,c_2
\end{equation}
with $\kappa$ the bending rigidity, $c_1$ and $c_2$ the (local) principal curvatures,  $c_0$ the spontaneous (or intrinsic) curvature, and $\kappa_\textrm{G}$ the Gaussian curvature modulus. 
%Helfrich's formula contains the physical principles underlying shape creation, that  must be universal for all cells.~\cite{zim-koz06}
Equation~\ref{eq:helfrich} can be used to calculate the elastic energy of a whole lipid bilayer: in this case the total and Gaussian curvatures describe the bilayer midsurface, and the elastic constant characterising the bilayer can be denoted by $\kappa^b$, $\kappa_G^b$ and $c_0^b$.
Helfrich's model can also describe the curvature energy of each of the two monolayers: in this case it is convenient to define the so-called neutral surface (identical to the pivotal surface where the interfacial curvature is not great~\cite{templer98}) of the monolayer, which is shifted by distance $\xi$ from the bilayer midplane toward the lipid-water interface. The monolayer elastic characteristics are denoted $\kappa^m$, $\kappa_G^m$ and $c_0^m$. For the case where the monolayer is held flat, the stored curvature elastic energy is simply $2\kappa^m (c_0^m)^2$. % templer 98
Lipid monolayers typically display a so-called {\em spontaneous curvature}. When a bilayer is made of monolayers with nonzero spontaneous curvature, it is affected by a built-in frustration defined as \index{curvature!stress field} {\em curvature stress field}. % mouritsen, p.49
If the bilayer cannot sustain the curvature stress, non-lammellar structures will form.


\subsubsection{Bending modulus}\index{bending modulus}
CHECK~\cite[2.6.1]{seifert95}, where a summary of exp techniques is given!!!!!
The bending modulus $\kappa$ for a flat surface is defined via the energy per unit area $E_\kappa$ that is required to produce a mean curvature $H$ of the surface:
\begin{equation}
E_\kappa = 2\kappa H^2
\end{equation}
where the mean curvature is given by the two principal curvature radii $R_1$ and $R_2$ as $H=(1/R_1+1/R_2)/2$.

\subsubsection{Gaussian curvature modulus}\index{gaussian curvature modulus}
The Gaussian curvature modulus $\kappa_G$ controls the topological complexity of the interface; $\kappa_G$ is difficult to determine experimentally, but it is believed to be of the same order of magnitude of the mean curvature modulus $\kappa$~\cite{mouritsen}. \cite{templer98b} proposed the relation $-1<\kappa_G/\kappa<0$.
For fluid phase DMPC, the Gaussian curvature is therefore bound to lie within the range $-7\,k_BT<\kappa_G<0$.

\subsubsection{Torque tension}\index{lateral pressure profile!torque tension}
At mechanical equilibrium, the summed lateral pressure ditribution must be zero, but the first moment, the torque tension $\tau$, is in general nonzero. The torque tension can be espressed as: 
\begin{equation}
\tau = \int_{0}^{h}z\,\pi(z)\:\de z
\end{equation}
where $z=0$ at the centre of the bilayer and $z=\pm h$ at the aqueous interfaces. The first moment of the lateral pressure $\tau$ is also called the {\em torque tension}, because it is the torque stored in a monolayer which is forced to remain flat~\cite{attard00}. The first moment is in essence a measure of a shift in the centre of the lateral pressure distribution in each monolayer away from ($\tau>0$) or toward ($\tau<0$) the bilayer centre~\cite{cantor01}. 
The first moment of the pressure profile gives the product of the splay curvature elastic modulus $k$ and the spontaneous curvature $c_o$ of the monolayer:
\begin{equation}
\tau = k c_0
\end{equation}
the value of $c_0$ being important in determining the ``non-lamellar'' tendency, i.e. the tendency of the lipid to form cylindircal aggregates such as inverted hexagonal phases. 
The torque tension of a monolayer is related to the curvature elastic parameters via 
\begin{equation}
\tau = 2\kappa c_0
\end{equation}
being $\kappa$ the bending rigidity and $c_0$ the monolayer spontaneous curvature. Since $c_0$ is positive for type-I micelle-forming lipids, and negative for type-II lipids (that wish to bend towards the water), the torque tension is positive for type-I lipids and negative for type-II lipids.

The elastic stress \index{elastic stress} changes the membrane's physical properties and manifests itself in at least two biologically relevant functional aspects~\cite{bezrukov00}:
\begin{itemize}
\item by modifying the energetics of hydrophobic inclusions, it influences protein-lipid interactions;
\item by changing the energetics of spontaneous formation of non-lamellar local structures, it influences membrane stability and fusion.
\end{itemize}

\subsubsection{Gaussian curvature modulus}\index{lateral pressure profile!Gaussian curvature modulus}
The Gaussian curvature modulus is given by the second moment of the lateral pressure profile~\cite{kozlov89, shearman06}:
\begin{equation}
\kappa_G = \int_{0}^{h}z^2\,\pi(z)\:\de z
\end{equation}
The Gaussian modulus $\kappa_G$ describes the energy required to change the Gaussian curvature\footnote{The Gaussian curvature $K$ is defined as $K=c_1c_2$, being $c_1$ and $c_2$ the principal curvatures in a point on a monolayer~\cite{shearman06}.} of a (monolayer) surface~\cite{shearman06}. 
\cite{attard00}
mechanosensitive channels~\cite{perozo02}


\subsubsection{Integral moments of the pressure profile}
At mechanical equilibrium, the summed lateral pressure ditribution must be zero. However, the {\em integral moments} of the lateral pressure distribution are in general nonzero, and they are directly related to fundamental curvature geometric and elastic parameters. We will compute the first and second integral moments of $\pi(z)$, which yield the {\em torque tension} and the {\em gaussian curvature modulus}. 

\paragraph{Torque tension and spontaneous curvature}
The torque tension $\tau$ of a monolayer (i.e. half the bilayer) is the first moment of the lateral pressure profile $P_1$:
\begin{equation}
\tau =P_1= \int_{0}^{h}z\,\pi(z)\:\de z
\end{equation}
\begin{equation}
\kappa c_0 = \int_{0}^{h}z\,\pi(z)\:\de z
\end{equation}
where $z=0$ at the centre of the bilayer and $z= h$ at the aqueous interface. 
 The first moment of the lateral pressure is called the {\em torque tension} because it is the torque stored in a monolayer which is forced to remain flat~\cite{attard00}. 
The torque tension $\tau$ of a monolayer is related to the curvature elastic parameters via: 
\begin{equation}\label{eq:taukc0}
\tau = \kappa c_0
\end{equation}
being $\kappa$ the bilayer bending rigidity (also called {\em splay curvature elastic modulus}) and $c_0$ the monolayer spontaneous curvature. Table~\ref{tab:spontCurvData} reports experimental estimates for a selection of lipid species, along with our value for DMPC computed from Equation~\ref{eq:taukc0} considering the experimental value for $\kappa_{DMPC}$.
\begin{table}
\begin{center}
%\vspace{8pt}
\begin{tabular}{lcl} % put @{} if you want to eliminate horizontal space between columns
%\hline
{\em Method~[reference]} & {\em Lipid} & $c_0$ / nm$^{-1}$ \\
Exp.~\cite{leikin96}  & DOPE & $-0.333$\\
Exp.~\cite{chen97,szule02}    & DOPC & $-0.050\div-0.115$\\
Exp.~\cite{fuller03}  & DOPS & $+0.069$ \\
Coarse-grain MD~[{\em this work}] & DMPC & 
\end{tabular}
\caption[Monolayer spontaneous curvatures]{Monolayer spontaneous curvatures. Collection of experimental data for a number of abundant lipid species. The last line reports our estimate from Molecular Dynamics calculation.}
\label{tab:spontCurvData}
\end{center}
\end{table}

\paragraph{Gaussian curvature modulus}
The Gaussian curvature modulus $\kappa_G$ of a monolayer derives from the first and second integral moments $P_1$ and $P_2$ of the lateral pressure profile via:
\begin{eqnarray}
P_2      =& \int_{0}^{h}z^2\,\pi(z)\:\de z& \\
\kappa_G =&  - \int_{0}^{h}(z-\xi)^2\,\pi(z)\:\de z =& 2\xi P_1 - P_2 
\end{eqnarray}
being $\xi$ the distance to the {\em pivotal surface}, defined as the surface at which there is no change in the molecular cross-sectional area upon bending~\cite{shearman06}. The pivotal surface \index{pivotal surface} has been experimentally identified close to the polar/apolar interface~\cite{rand90,chung94,templer95,chen97}: we therefore set $\xi=D_C$, considering the experimental hydrophobic thickness $2D_C=2.54$\,nm of fluid-phase DMPC bilayers~\cite{kucer05a}. 
As for the first integral moment, $z=0$ at the centre of the bilayer and $z= h$ at the aqueous interfaces. 
The gaussian curvature modulus describes the energy required to change the gaussian curvature  of a (monolayer) surface~\cite{kozlov89, shearman06}. The Gaussian curvature $K$ is defined as $K=c_1c_2$, being $c_1$ and $c_2$ the principal curvatures in a point on a monolayer. The Gaussian curvature modulus $\kappa_G$ is then related to the general expression for the bending energy through equation~\ref{eq:helf}.
The value of the monolayer Gaussian curvature modulus, which is difficult to determine experimentally, is predicted to be negative from theoretical arguments. In fact, \cite{templer98b} have shown that the monolayer Gaussian curvature modulus is bound to the bending rigidity according to %\footnote{Note that our convention for the sign of the lateral pressure profile is opposite to that of~\cite{templer98}, hence the gaussian curvature modulus signs are also opposite.} 
$-1 \le\kappa_G^m/\kappa^m\le 0$: this prediction has been confirmed by the (few) available experimental data~\cite[Table~3]{shearman06}.  
\cite{marsh06} has observed that experimental data suggest the relation $\kappa_G\approx-0.8\kappa$ for lipid monolayers in general.  

\cite{cantor99b} predicted the first and second moments of the pressure profile for the hydrocarbon region of 14:0 lipid bilayers to be $P_1=-1.74\,k_BT/$\AA, and  $P_2=-29.4\,k_BT$. These results cannot be representative of a complete DMPC bilayer as they do not take into account headgroups and hydrating water.

\subsubsection{Biophysical importance of the lateral pressure profile}
 The intrinsic curvature $c_0$ minimizes the bending energy of a lipid monolayer, and quantitatively gives the tendency to curl~\cite{gruner85}. The value of $c_0$ also relates to the tendency of the lipid to aggregate into non-lamellar structures. 
\cite{attard00} showed how both the sign and the magnitude of the torque tension play a key role in controlling membrane lipid synthesis. It seems reasonable to argue that the stored curvature elastic energy, via the integral moments of the lateral pressure distribution, generally regulates the activity of membrane proteins. This purely physical mechanism is simple and robust, and it ensures membrane's integrity. There is in fact growing evidence about the plausibility of this theory~\cite{gruner85,gruner94,curnow04,lundbaek04,booth05}.   

\subsection{Water permeation}

Experimental data for the permeability coefficient of water through phospholipid bilayers are collected in Tab.~\ref{tab:watPermData}. 

\begin{table}[]
\begin{center}
\vspace{8pt}
\begin{tabular}{lcc} % put @{} if you want to eliminate horizontal space between columns
{\em Lipid} & $P_\textrm{W}$ [$\mu$m/s]  & {\em Reference }  \\
diC(14:0)PtdCho (DMPC) & 70 & \cite[fig.~17b]{bloom91}\\
diC(16:0)PtdCho (DPPC) & 24 & \cite[p.~96]{cevc}\\
diC(14:1)PtdCho &  240  &  \cite{paula96}\\
diC(18:1c$\Delta^9$)PtdCho (DOPC) & 42 & \cite{olbrich00}\\
diC(18:1c$\Delta^9$)PtdCho (DOPC) & 97 & \cite{huster97}\\
Egg lecithin & 2  &  \cite{carruthers83}\\
\end{tabular}
\caption[Experimental water permeability]{Experimental water permeability coefficient. Measurements taken at full hydration in the L$_\alpha$ phase.}
\label{tab:watPermData}
\end{center}
\end{table}

%Also, \cite{olbrich00} measured the water permeability of polyunsaturated lipid bilayers: the apparent coefficient for water permeability, at 21\textcelsius, varied modestly in a range from approximately 30 to 40\,$\mu$m/s for mono- and dimono-unsaturated PCs.
 

\section[The matrix of life]{The matrix of life: weak interactions in an aqueous environment}

Table~\ref{tab:elInt}. 

\begin{table}\centering
\begin{tabular}{lc}
Type & Dependence of energy on distance\smallskip\\
Charge-charge & $1/r$\\
Charge-dipole & $1/r^2$\\
Dipole-dipole & $1/r^3$
\end{tabular}
\caption[Electrostatic interactions]{Electrostatic interactions.}\label{tab:elInt}
\end{table}


\subsection{Charge-charge interactions}

Many of the molecules present in cells, including macromolecules like DNA or proteins, carry a net electrical charge. At the same time, the cell contains an abundance of small ions, like Na$^+$ and Cl$^-$.
The force between a pair of charges $Q_i$ and $Q_j$, separated in a vacuum by a distance $r$, is given by Coulomb's law:
\begin{equation}\label{eq:coulForce}
F(r)=k\frac{Q_iQ_j}{r^2}
\end{equation}
where $k$ is a constant whose value depends on the units used.
A fundamental problem with equation~(\ref{eq:coulForce}) as it stands is that the biological environment is never a vacuum.
The charges are always separated by water or other molecules or parts of molecules. The existence of such a {\em dielectric medium} between charges has the effetc of screening them from one another, so that the actual force is always less than that given by equation~(\ref{eq:coulForce}).
This screening effect is expressed by inserting a dimensionless number, the {\em dielectric constant} $\epsilon$, in equation~(\ref{eq:coulForce}):
\begin{equation}\label{eq:coulForceScreened}
F(r)=k\frac{Q_iQ_j}{\epsilon \,r^2}
\end{equation}
The dielectric constant of water is very high, approximately 80, whereas organic substances usually have much lower values, in the range 1-10. The major consequence of this large value in water is obvious: charged particles like ions interact rather weakly in an aqueous environment unless they are very close to one another~\cite{mathews}.

\subsection{Interactions involving permanent dipoles}

Some molecules carry no net charge but have an asymmetric internal distribution of charge. 
For instance, in the water molecule the electron distribution is such that the end towards the oxygen is slightly negative and the end toward the two hydrogen atoms is slightly positive. Such a molecule is a {\em dipole}, with permanent dipole moment $\mu$. In particular, if a molecule has fractional charges $+q$ and $-q$, separated by a distance $x$, the dipole moment is a vector whose magnitude is:
\begin{equation}\label{eq:dipMag}
\mu=qx
\end{equation}
The common units of dipole moment are {\em debyes}~(D), being 1\,D equal to $3.34\times10^{-30}$\:C\,m. 
The water dipole moment in the liquid phase has an average value of about 3\,D, as reported by~\cite{silve99a, gubsk02a}.

Perhaps the most interesting property of phospholipids is their large dipole moment. The tendency to orient their dipoles parallel to the surface, which is favourable in terms of the free energy of electrostatic interactions, and the larger degree of motional freedom in the water phase and steric orientation, are expected to result in a wide distribution of headgroup tilt angles~\cite{tiele97a}.
Indeed, the conformation of the polar head group of lipids in the liquid-crystalline state (L$_\alpha$) has been experimentally deduced: the P-N vector makes an angle between 60 and $90^{\circ}$ with the membrane normal, i.e. it lies approximately parallel, within $30^{\circ}$, to the membrane plane, slightly pointing towards the water phase\footnote{Similar results have been obtained for all major classes of phospholipids (PGs, PCs, PEs), for temperatures well into the (fluid lamellar) $L_\alpha$ phase; hence, this behaviour has been considered a common feature of lipid headgroup dipoles in the $L_\alpha$ phase~\cite{saizl02abis}.}~\cite{akuts91a, sch89}.
